In Crystal Field Theory what does Δo=10Dq mean?

Question

While studying the Crystal Field Theory I was told $\mathrm{Dq}$ is a unit, related to the unit $\Delta_{\mathrm{O}}$ by the relation $\Delta_{\mathrm{O}} = 10\ \mathrm{Dq}$. But aren't $\Delta_{\mathrm{O}}$ and $\mathrm{Dq}$ variables, not units? The unit is of energy, such as $\mathrm{eV}$ or Joules, these are symbols to represent a particular value of energy, which could be anything. So what do we mean when we say $\Delta_{\mathrm{O}}= 10\ \mathrm{Dq}$? What exactly is $\mathrm{Dq}$ (I haven't been able to find this in any of my textbooks, they just either state this relation, or not at all) and are they actually units of energy?

Answer 1

Though I don't think I can provide a full answer, I did see this in a lecture series. $\Delta_{\mathrm{O}}$ is the ligand field splitting and it is a measure for the strong or weak field that the ligands of a complex create. Depending on the ligands the energy levels of, for example octahedral iron complexes (below), are closer to each other in a low field and farther apart for a higher field. This is dependant on $\Delta_{\mathrm{O}}$. This is also an energy difference and the units are in joules or electron volts. See the picture below. You might also want to look up Tanabe-Sugano Diagrams.

I was told (sorry, no proper reference) that the expression $\Delta_{\mathrm{O}}= \pu{10Dq}$ comes from quantum mechanics. The reason that this term is simply stated instead of defined, lies in the origin of CFT. The theory was largely empirical and thus $\Delta_{\mathrm{O}}$ was used. This difference in energy levels could also be described in quantum systems and is called $\pu{Dq}$ after its quantum number. The difference between the levels appeared to be $\pu{10Dq}$.

I hope this is not a gross oversimplification, but a rather helpful one!

Answer 2

The origin of the usage of $\pu{Dq}$ is discussed elsewhere and is out of scope for this question; but I can tell you that it is indeed a relative unit.

When looking at complexes without knowing their specific electronic structure and specific energy levels, the $\mathrm{Dq}$-expression is a simple way of comparing the relative electron stabilisations. The idea is that the entire field split is given as $10~\mathrm{Dq}$ and that a hypothetical zero-level is introduced such that one electron in each orbital would give an overall stabilisation of $0~\mathrm{Dq}$. So in the case of a $\mathrm{d^3}$ complex, the ‘stabilisation’ would be $-18~\mathrm{Dq}$.

The convention has a number of flaws:

• As noted, the overall energy can differ greatly. No two complexes will have exactly the same value for $10~\mathrm{Dq}$.

• The value of $10~\mathrm{Dq}$ even depends strongly on the geometry: octahedral complexes have a much larger value of $10~\mathrm{Dq}$ than tetrahedral ones:

$$4.44~\mathrm{Dq}(\text{oct}) \approx 10~\mathrm{Dq}(\text{tet})$$

• The notation does not extend well to complexes in geometries other than octahedral or tetrahedral. For Jahn-Teller distorted or square-planar complexes, the definition of $10~\mathrm{Dq}$ becomes unclear and less useful. It is often given as the energy difference corresponding to the highest-wavelength electronic absorption.

So overall, while it is helpful to use it at a basic level, it outlives its usefulness quickly due to being, as you said, relative.